Topological Structures of the Alpha Group via Monte Carlo Simulations

We present a Monte Carlo dynamical simulation of the Alpha Group, a novel algebraic–topological framework in which the geometry of space emerges from the structure of the operator, M(θ), defined through the imaginary component, μ, and a projective division mechanism. The operator encodes two nontrivial nodal structures that influence the interaction dynamics and resulting topology. By evolving a normalized four-dimensional state vector through repeated iterations of M(θ) with small stochastic perturbations, we obtain a trajectory that reveals the intrinsic structure of the underlying space. The Monte Carlo simulation is performed over 80,000 steps near θ=π/2, producing a high-resolution dataset corresponding to an emergent three-dimensional subspace. The resulting trajectory is analyzed using graph-based methods. Three-dimensional visualizations in transverse planes and two-dimensional projections (XY, XZ, YZ) are used to identify structural centers, ranges, and statistical characteristics. Topological features are extracted via proximity graphs, where 3-cliques (triangles) define 2-simplices, and homology is computed using the Euler–Poincaré relation χ = V − E +F = β0 − β1+ β2. The results show a single connected component (β0=1), the presence of nontrivial one-dimensional cycles (β1>0), and persistent two-dimensional cavities (β2>0). The Euler characteristic is consistent, confirming internal topological coherence. The Monte Carlo analysis provides computational evidence that the emergent geometry of the Alpha Group is non-Riemannian and dynamically generated, with a homological structure influenced by the imaginary sector μ. These findings suggest that space may exhibit a dynamic, parameter-dependent topology, opening new directions for mathematical and physical investigations of complex geometric structures.